在显示模式下，给出的数学公式如下：

二元样本回归方程为
$$\boldsymbol{Y}=\boldsymbol{X} \hat{\boldsymbol{\beta}}^{\prime}+e$$

参数的估计值为
$$\hat{\boldsymbol{\beta}}=\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1}\left(\boldsymbol{X}^{\prime} \boldsymbol{Y}\right)$$

于是
$$\left(\begin{array}{l}
\hat{\beta}_{0} \\
\hat{\beta}_{1} \\
\hat{\beta}_{2}
\end{array}\right)=\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1}\left(\boldsymbol{X}^{\prime} \boldsymbol{Y}\right)=\left(\begin{array}{c}
626.509 \\
-9.79057 \\
0.02862
\end{array}\right)$$

根据随机干扰项方差的估计式 $\hat{\sigma}^{2}=\frac{\sum e_{i}^{2}}{n-k-1}$ 得到
$$\hat{\sigma}^{2}=\frac{e^{\prime} \boldsymbol{e}}{n-k-1}$$

而
$$\begin{aligned}
\boldsymbol{e}^{\prime} \boldsymbol{e} & =(\boldsymbol{Y}-\hat{\boldsymbol{Y}})^{\prime}(\boldsymbol{Y}-\hat{\boldsymbol{Y}})=(\boldsymbol{Y}-\boldsymbol{X} \hat{\boldsymbol{\beta}})^{\prime}(\boldsymbol{Y}-\boldsymbol{X} \hat{\boldsymbol{\beta}}) \\
& =\boldsymbol{Y}^{\prime} \boldsymbol{Y}-\boldsymbol{Y}^{\prime} \boldsymbol{X} \hat{\boldsymbol{\beta}}-\hat{\boldsymbol{\beta}}^{\prime} \boldsymbol{X}^{\prime} \boldsymbol{Y}+\hat{\boldsymbol{\beta}}^{\prime} \boldsymbol{X}^{\prime} \boldsymbol{X} \hat{\boldsymbol{\beta}} \\
& =\boldsymbol{Y}^{\prime} \boldsymbol{Y}-\boldsymbol{Y}^{\prime} \boldsymbol{X} \hat{\boldsymbol{\beta}}-\hat{\boldsymbol{\beta}}^{\prime} \boldsymbol{X}^{\prime} \boldsymbol{Y}+\hat{\boldsymbol{\beta}}^{\prime} \boldsymbol{X}^{\prime} \boldsymbol{X}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{Y} \\
& =\boldsymbol{Y}^{\prime} \boldsymbol{Y}-\boldsymbol{Y}^{\prime} \boldsymbol{X} \hat{\boldsymbol{\beta}} \\
& =4515071.83-4512954.98=2116.85
\end{aligned}$$

故
$$\hat{\sigma}^{2}=\frac{e^{\prime} e}{n-k-1}=\frac{2116.85}{10-2-1}=302.41$$

又由于
$$\begin{array}{l}
\mathrm{TSS}=\sum\left(Y_{i}-\bar{Y}\right)^{2}=\sum\left(Y_{i}^{2}-2 \bar{Y} Y_{i}+\bar{Y}^{2}\right) \\
=\sum Y_{i}^{2}-n \bar{Y}^{2}=\boldsymbol{Y}^{\prime} \boldsymbol{Y}-n \bar{Y}^{2} \\
=4515071.83-10 \times 449342.31=21648.73
\end{array}$$